Computer-based simulation enables the analysis of complex systems before the time and expense of actually implementing the systems is incurred. In general, a simulator is a software application that analyzes a system (e.g., a circuit design) based upon a description of that system to predict the behavior of the system under a given set of conditions or assumptions. Typically, the system description is made up of equations describing the desired behavior of the system. The simulator solves these equations in the context of simulated conditions represented by stimuli applied to the system in order to simulate operation of the system under those conditions.
A circuit simulator typically constructs circuit equations from mathematical models of the components in a circuit. The mathematical models may be provided in the circuit simulator (e.g., the semiconductor device models of SPICE-like simulators), and/or may be specified by a user using a hardware description language (e.g., Verilog-AMS, Verilog-A). The circuit simulator combines the mathematical models of the components with equations that describe the interconnection of the components to construct a set of nonlinear differential algebraic equations (DAEs) that describe the circuit. In many circuit simulators, the interconnection equations are derived from Kirchoff's voltage and current laws which require that the sum of all currents flowing out of a node at any instant is zero and the algebraic sum of all branch voltages around a loop at any instant is zero.
A circuit simulator may be used to perform many different types of analysis on a circuit, e.g., transient analysis, DC analysis, AC analysis, harmonic balance analysis, Monte Carlo analysis, mismatch analysis and parametric analysis, corners analysis, etc. For a transient analysis, the circuit simulator computes the response of a circuit as a function of time. To perform the transient analysis, the differential algebraic equations (DAEs) describing the circuit are discretized in time to convert the solution of the equations into a problem of solving a sequence of systems of nonlinear algebraic equations (NAEs). In other words, the differential equations are numerically integrated to replace the time derivative operator with a discrete-time approximation and the resulting finite-difference equations are solved one time point at a time starting from some initial condition. Four integration methods commonly used in circuit simulation include trapezoidal rule, forward Euler, backward Euler, and backward difference formulas (i.e., Gear's methods).
More specifically, transient analysis determines the time domain response of the circuit over a time interval [0, T]. The transient solution to the system of differential equations is computed by dividing this time interval into discrete time points [0, t1, t2, . . . , T], and applying a numerical integration algorithm at each time point to transform the differential equations into equivalent algebraic equations. The circuit simulator solves these nonlinear algebraic equations (NAEs) on every time point of a transient analysis. The solution to the NAEs is obtained by executing an iterative sequence of linearized solutions. The Newton-Raphson method, i.e., the Newton method or Newton iteration, is the most common method of linearization used. The Newton-Raphson method begins with an initial guess for the solution to the NAEs. The NAEs are then linearized about that guess, and the resulting linear system is solved. One common technique used to solve such a linear system in circuit simulators is LU factorization. LU factorization is a modified version of Gaussian elimination. The linear equations of the linear system are represented as a circuit matrix, i.e., a modified nodal analysis (MNA) circuit matrix which is typically a Jacobian matrix, and the matrix is factorized into a product of lower- and upper-triangular matrices. Then, forward and backward substitutions are performed to obtain the solution. Once the solution is obtained, the NAEs are then re-linearized about the solution and the procedure repeats until the process converges.
Threshold-crossing events (i.e., cross events) are widely used in transient analysis. In general, a cross event is an event that is fired during simulation whenever the result of evaluating an expression crosses through a threshold (e.g., 0) in a positive and/or negative direction. In essence, a cross event is specified to cause the simulator to generate a time point, i.e., a crossing time point, at the threshold crossing point, i.e., to cause the simulator to solve the circuit equations at the threshold crossing point. For example, the cross function in Verilog-A generates a monitored analog event to detect threshold crossing and is also used in Verilog-AMS for threshold crossing.
One frequently used approach for detecting and resolving a crossing time point during transient analysis is as follows. At each time point, the circuit simulator checks for all cross conditions. If violation of a cross condition is detected (e.g., the result of evaluating an expression crosses through a threshold), the circuit simulator rejects the time point and enters a search mode to attempt to find a time point where the crossing that violated the cross condition is occurring. The circuit simulator iteratively predicts (via interpolation) and solves the crossing point expression(s) until a time point sufficiently close (i.e., within some error tolerance) to the actual time at which the crossing occurs is found. The iterative prediction is typically repeated for 2-4 time points to locate an accurate crossing time point and may require many more time points. Further, the entire circuit or system of equations is evaluated and solved for each of these time points, which is computationally expensive. Therefore, specifying a large number of cross events may significantly impact the performance of a simulation. Accordingly, improvements in the processing of threshold-crossing events are desirable.